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2008
When we have a large number of independent replications of a random experiment, we observe that the frequency of the outcomes can be very well approximated by the probabilities of the corresponding events. The profits of many commercially successful enterprises — like casinos or insurance companies — are based on random events obeying some laws.
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When we have a large number of independent replications of a random experiment, we observe that the frequency of the outcomes can be very well approximated by the probabilities of the corresponding events. The profits of many commercially successful enterprises — like casinos or insurance companies — are based on random events obeying some laws.
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Bernoulli's Law of Large Numbers and the Strong Law of Large Numbers
Theory of Probability & Its Applications, 2016We discuss the connection between the law of large numbers and the strong law of large number for sums of independent random variables. The mathematical content of these laws and their connection with mathematical statistics problems is compared.
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Aggregation and the law of large numbers in large economies
Games and Economic Behavior, 2004zbMATH Open Web Interface contents unavailable due to conflicting licenses.
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2012
We again turn our discussion to sequences of independent random variables. Let \({\xi }_{1},{\xi }_{2},\ldots \) be a sequence of random variables with finite expectations \({m}_{n} =\mathrm{ E}{\xi }_{n}\), \(n = 1,2,\ldots \). Let \({\zeta }_{n} = ({\xi }_{1} + \ldots + {\xi }_{n})/n\) and \({\overline{\zeta }}_{n} = ({m}_{1} + \ldots + {m}_{n})/n\).
Leonid Koralov, Yakov G. Sinai
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We again turn our discussion to sequences of independent random variables. Let \({\xi }_{1},{\xi }_{2},\ldots \) be a sequence of random variables with finite expectations \({m}_{n} =\mathrm{ E}{\xi }_{n}\), \(n = 1,2,\ldots \). Let \({\zeta }_{n} = ({\xi }_{1} + \ldots + {\xi }_{n})/n\) and \({\overline{\zeta }}_{n} = ({m}_{1} + \ldots + {m}_{n})/n\).
Leonid Koralov, Yakov G. Sinai
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Constitutional Commentary, 2002
Book review ; Can mathematics be used to inform legal analysis? This is not a ridiculous question. Law has certain superficial resemblances to mathematics. One might view the Constitution and various statutes as providing "axioms" for a deductive legal system.
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Book review ; Can mathematics be used to inform legal analysis? This is not a ridiculous question. Law has certain superficial resemblances to mathematics. One might view the Constitution and various statutes as providing "axioms" for a deductive legal system.
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1994
Assume that m = 1 and \(E\left| {\Phi ({X_1})} \right| < \infty \) in (1.1.3). Then, according to the classical Kolmogorov law of large numbers for the sums of independent random variables, we have U n → EΦ(X 1) as n → ∞ with probability 1. Let us show that this statement is valid for U-statistics of an arbitrary degree m ≥ 1.
V. S. Koroljuk, Yu. V. Borovskich
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Assume that m = 1 and \(E\left| {\Phi ({X_1})} \right| < \infty \) in (1.1.3). Then, according to the classical Kolmogorov law of large numbers for the sums of independent random variables, we have U n → EΦ(X 1) as n → ∞ with probability 1. Let us show that this statement is valid for U-statistics of an arbitrary degree m ≥ 1.
V. S. Koroljuk, Yu. V. Borovskich
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2000
One of the fundamental results of Probability Theory is the Strong Law of Large Numbers. It helps to justify our intuitive notions of what probability actually is (Example 1), and it has many direct applications, such as (for example) Monte Carlo estimation theory (see Example 2).
Jean Jacod, Philip Protter
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One of the fundamental results of Probability Theory is the Strong Law of Large Numbers. It helps to justify our intuitive notions of what probability actually is (Example 1), and it has many direct applications, such as (for example) Monte Carlo estimation theory (see Example 2).
Jean Jacod, Philip Protter
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2003
Let X1, ..., X n be a random sample from N(μ, σ2) and define \( % MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr ...
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Let X1, ..., X n be a random sample from N(μ, σ2) and define \( % MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr ...
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2015
A sequence of random variables \(X_{1}, X_{2}, \ldots ,\) is said to converge in probability \(^1\) to zero, if the probability of the relation \(|X_{n}| > \epsilon > 0\) tends to zero as \(t\) increases, i.e. if the distribution functions \(V_{n}(x)\) of the \(X_{n}\) satisfy.
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A sequence of random variables \(X_{1}, X_{2}, \ldots ,\) is said to converge in probability \(^1\) to zero, if the probability of the relation \(|X_{n}| > \epsilon > 0\) tends to zero as \(t\) increases, i.e. if the distribution functions \(V_{n}(x)\) of the \(X_{n}\) satisfy.
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1975
We shall consider a sequence of random variables {X n ; n = 1, 2, ...}. By definition, this sequence converges in probability to the random variable X if P(│X n − X│ ≧ e) → 0 for every fixed e > 0. In this case we use the notation \({X_n}\;\xrightarrow{P}\;X\).
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We shall consider a sequence of random variables {X n ; n = 1, 2, ...}. By definition, this sequence converges in probability to the random variable X if P(│X n − X│ ≧ e) → 0 for every fixed e > 0. In this case we use the notation \({X_n}\;\xrightarrow{P}\;X\).
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